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33

The minimum variance solution loads up on securities that have low variances and co-variances. Theoretically you are correct that this should have a low expected return profile. However, it turns out - in contradiction to modern portfolio theory - that securities that have low-volatility or low-beta experience higher returns than high-volatility or high-...


25

Yes, the weights of the first eigenvector of a covariance matrix represent the market factor and also the largest source of systematic risk (variation of returns). Why PCA? Well, PCA simply identifies the eigenvector that maximally explains the variance of the system. It turns out that this is the "market factor" - i.e. the tendency of securities to rise ...


15

The following papers may help. A New Look at Minimum Variance Investing by Bernd Scherer Minimum Variance Portfolio Composition by Clarke, De Silva & Thorley Under a multifactor risk-based model, if the global minimum variance portfolio dominates the market portfolio, the implication is that the market portfolio is not multifactor efficient and that ...


13

Statistically you would apply Bessel's correction to address the bias you point out. However, that misses the point that the variance-covariance matrix is non-stationary, suffers from the curse of dimensionality, and that the noisy mean return estimates have significantly more impact than a biased covariance matrix on portfolio weights. The best ways to ...


12

Unlike the tangency portfolio on the efficient frontier (which represents the most efficient portfolio in terms of max expected sharp ratio), min var portfolios have no ex ante theory that suggests it should outperform a cap weighted market portfolio. The same can be said about other risk-weighted portfolio construction schemes, including equal risk ...


12

I'm just providing a global answer to the question, as I think it can be interesting for some beginners in quant finance. The properties given by TheBridge: Normalize $\rho (\emptyset)=0$ This means you have no risk in taking no position. Sub-addiitivity $\rho(A_1+A_2) \leq \rho(A_1)+\rho(A_2)$ Having a position in two different can only decrease the ...


12

The unconstrained mean-variance problem $$w_{mv,unc}\equiv argmax\left\{ w'\mu-\frac{1}{2}\lambda w'\Sigma w\right\} $$ can easily be found by taking the derivative $$\frac{\partial}{\partial w}\left(w'\mu-\frac{1}{2}\lambda w'\Sigma w\right)=\mu-\lambda\Sigma w $$ setting it to zero, and solving for $w$. This gives $$w_{mv,unc}\equiv\frac{1}{\lambda}\...


11

After having done a lot of research on the topic I found the following excellent research piece on ETF.com: Wealthfront modifies historic asset-class returns with current market implied expected returns (Black-Litterman) as well as with the in-house views of Chief Investment Officer Burton Malkiel’s team. In addition, Wealthfront sets minimum and ...


9

The minimum variance optimization framework does not guarantee positive return whatsoever. As a matter of fact what you are trying to do is something close to the following: $$\underset{w}{\arg \min} \quad w' Q w \quad \text{s.t} \quad Aw \leq b,\quad \sum_i w_i=1$$ The fact that you get positive return is a nice result that you get from your backtest (i....


9

Yes. Check out the Lower Partial Moments literature. In my view the best introduction to this is Narwrocki - A Brief History of Downside Risk Measures. Uryasev established equivalence between CVaR approach and low partial moments. If Markowitz had the tools at the time time, LPM utility functions would be the introductory optimization model as opposed to ...


8

Tools from the field of stochastic optimization are best suited for these problems. In particular, attached is a paper on non-parametric density estimation for stochastic optimization that describes an algorithm if state variables can be associated with draws from the predictive distribution. Here's another approach by Kuhn. These are all one-period ...


8

Bernd Scherer has done exactly this test in his text "Portfolio Construction and Risk Budgeting 4th Edition". There is an SSRN paper by Scherer called "Resampled Efficiency and Portfolio Choice (2004)" you can take a look at as well. I would suggest you skip re-sampling (especially if you have a long-only portfolio) and take a look at Meucci's Robot ...


8

The initial investment is the capital in the account used to support the portfolio, not the cost of the assets in the portfolio. For example, when you sell a stock or bond short, your account doesn't actually accrue any cash. Instead you start receiving a regular cash flow. There isn't necessarily a difference between these quantities in a long-only ...


8

Portfolio optimalisation depends heavily on the estimation of the moments (and therefore has HUGE estimation uncertainty). Even though it's useful for comparing and analysing different existing strategies, I think practitioners are moving more towards the usage of factor portfolios for the strategies themselves (e.g. Fama-French). Also because the ...


8

Alphas from a time-series regression are error terms in the cross-sectional, linear relationship between expected returns and factor betas. If a factor model were correct those error terms (the alphas) would be zero. Discussion A carefully written version of a standard time-series regression of returns in excess of the risk free rate on market excess ...


8

The best explanation I have seen so far is the so-called Adaptive Market Hypothesis by Andrew Lo: The adaptive market hypothesis, as proposed by Andrew Lo, is an attempt to reconcile economic theories based on the efficient market hypothesis (which implies that markets are efficient) with behavioral economics, by applying the principles of ...


7

The term in sample and out of sample are commonly used in any kind of optimization or fitting methods (MVO is just a particular case). When you make the optimization, you compute optimal parameters (usually the weights of the optimal portfolio in asset allocation) over a given data sample, for example, the returns of the securities of the portfolio for the ...


7

Check out following link. In page 23 you'll find the derivation. http://faculty.washington.edu/ezivot/econ424/portfolioTheoryMatrix.pdf


7

This is indeed a subtle point. What is generally meant with this statement is that correlation is going up in bear markets, so it is not so much the "turmoil" part (i.e. volatility per se) but the "trend" (i.e. negative in this case) part. Putting it another way is that when you control for volatility not the correlation but the covariance (which is the part ...


7

To clarify notation, you have an universe of $n=2000 \space$ stocks and two portfolio vectors $\mathbf{a},\mathbf{b}\in\mathbb{R}^{n}$ with $\left\|\mathbf{a}\right\|_{1}=\left\|\mathbf{b}\right\|_{1}=1$. Further, you have Estimators for the true Variance $\operatorname{Var}\left[\mathbf{a}\right]$ resp. $\operatorname{Var}\left[\mathbf{b}\right]$ and the ...


7

As a practitioner, I have worked on the following Maximize Yield/OAS for a Fixed Income Portfolio keeping the Rates Duration (Key Rate Durations) and Spread duration in a constrained range . There are other constraints such as No short selling Max amount you can buy is X% of Max outstanding amount in market Maximum exposure to a perticular country , ...


7

if you take the variance of a single asset it scales as a quadratic, $$ var(\lambda X) = \lambda^2 var(X) $$ so it's not surprising that the general case gives a quadratic form.


7

A lot has happened since Markowitz and Sharpe. While their work is still considered foundational, the empirical/practical relevance of their models has been questioned by later work. Here are a few more recent articles about portfolio theory, in no particular order (all accessible online): Jorion: Bayes-Stein Estimation for Portfolio Analysis, JFQA, 1986 ...


6

If you look in the portfolio management sections of the CFA (chartered financial analyst) curriculum, you'll find a listing of commonly used portfolio management techniques. It is by no means exhaustive, but the content in the CFA curriculum comes directly from industry professionals, so it is reasonable current and applicable. CFA Candidate Body of ...


6

The blog post http://www.portfolioprobe.com/2011/10/03/predictability-of-kurtosis-and-skewness-in-sp-constituents/ suggests that there is some predictability in kurtosis, but it isn't clear (to me at least) that there is enough predictabiilty to be useful. If there is a place for higher moments, my guess is that it is in asset allocation problems where ...


6

Mean-variance (MV) is a framework rather than a prescription. This framework allows one to make, discuss, and defend his investment decision. In practice, there are many ways to make adjustments to this framework, if you believe they will improve performance. E.g. you can adjust the framework by stating "I will MV-optimize weights subject to "0" if the ...


5

Adding a bit to the references mentioned by Quant Guy - apart from the aforementioned paper by Keating and Shadwick, Kazemi et al. introduce an alternative formulation of the Omega ratio (Sharpe-Omega) similar to the Sharpe ratio. As noted by Patrick Burns, higher moments could have some use when instruments other than equity are involved (hedge fund ...


5

There are many portfolio optimization paradigms that include a preference for skewness. These are generally alternatives meant to replace the modern portfolio management mean-variance framework developed by Markowitz. Skewness (or, more generally, higher moments) are only relevant in portfolio optimization if (a) assets are not normally distributed, and (b)...


5

The general optimization problem for portfolio management is the following: $$ \min x Q x $$ where $x$ is the allocation vector of your problem, and $Q$ is the covariance matrix of all your possible investments. In your example, you can compute the expectation $E[x]$, the variance $Var[x]=E[x^2]-E[x]^2$ and the covariane $cov[x,y]=E[xy]-E[x]E[y]$ pretty ...


5

Any explanations? Yes. Within each asset category we find that stocks may be: Unattractively underperforming the category norm Attractive as they meet the expected norm Unsustainable as their returns exceed the category norm and may suffer mean reversion By focusing on low variance, we exclude type (3) stocks that damage portfolio performance through high ...


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